Event Topology Classifiers at the Large Hadron Collider

Event classifiers are the most fundamental observables to probe the event topology of hadronic and nuclear collisions at relativistic energies. Over the last five decades, significant progress has been made to establish suitable event classifiers to probe different physics processes occurring in elementary $e^{+}e^{-}$ to heavy-ion collisions in a broad range of center of mass energies. One of the major motivations to revisit event classifiers at the Large Hadron Collider (LHC) originates from the recent measurements of high multiplicity proton-proton collisions, which have revealed that these small collision systems exhibit features similar to the formation of quark-gluon plasma (QGP), traditionally believed to be only achievable in heavy nucleus-nucleus collisions at ultra-relativistic energies. To pinpoint the origin of these QGP-like phenomena with substantially reduced autocorrelation and selection biases, and to bring all collision systems on equal footing, along with charged-particle multiplicity, lately several event topology classifiers such as transverse sphericity, transverse spherocity, relative transverse activity classifier, and charged-particle flattenicity have been used extensively in experiments as well as in the phenomenological front. In addition, the infrared and collinear safety of event-shape observables makes them ideal for precision studies of jets and heavy-flavors at the LHC. In this review article, we summarise the motivation, scope, and practical use of these event-shape observables. The discussion integrates results and insights from all major LHC experiments, setting the stage for precision investigations for Run 3, Run 4, and future high luminosity upgrades of the LHC.